An unbiased estimator that achieves this bound is said to be (fully) efficient. Such a solution achieves the lowest possible mean squared error among all unbiased methods, and is, therefore, the minimum variance unbiased (MVU) estimator. However, in some cases, no unbiased technique exists which achieves the bound. This may occur either if for any unbiased estimator, there exists another with a strictly smaller variance, or if an MVU estimator exists, but its variance is strictly greater than the inverse of the Fisher information.
The Cramér–Rao bound can also be used to bound the variance of biased estimators of given bias. In some cases, a biased approach can result in both a variance and a mean squared error that are below the unbiased Cramér–Rao lower bound; see estimator bias.
The Cramér–Rao bound is stated in this section for several increasingly general cases, beginning with the case in which the parameter is a scalar and its estimator is unbiased. All versions of the bound require certain regularity conditions, which hold for most well-behaved distributions. These conditions are listed later in this section.
Scalar unbiased case
Suppose is an unknown deterministic parameter that is to be estimated from independent observations (measurements) of , each from a distribution according to some probability density function. The variance of any unbiased estimator of is then bounded[12] by the reciprocal of the Fisher information:
where the Fisher information is defined by
and is the natural logarithm of the likelihood function for a single sample and denotes the expected value with respect to the density of . If not indicated, in what follows, the expectation is taken with respect to .
If is twice differentiable and certain regularity conditions hold, then the Fisher information can also be defined as follows:[13]
The efficiency of an unbiased estimator measures how close this estimator's variance comes to this lower bound; estimator efficiency is defined as
or the minimum possible variance for an unbiased estimator divided by its actual variance.
The Cramér–Rao lower bound thus gives
.
General scalar case
A more general form of the bound can be obtained by considering a biased estimator , whose expectation is not but a function of this parameter, say, . Hence is not generally equal to 0. In this case, the bound is given by
where is the derivative of (by ), and is the Fisher information defined above.
Bound on the variance of biased estimators
Apart from being a bound on estimators of functions of the parameter, this approach can be used to derive a bound on the variance of biased estimators with a given bias, as follows.[14] Consider an estimator with bias , and let . By the result above, any unbiased estimator whose expectation is has variance greater than or equal to . Thus, any estimator whose bias is given by a function satisfies[15]
The unbiased version of the bound is a special case of this result, with .
It's trivial to have a small variance − an "estimator" that is constant has a variance of zero. But from the above equation, we find that the mean squared error of a biased estimator is bounded by
using the standard decomposition of the MSE. Note, however, that if this bound might be less than the unbiased Cramér–Rao bound . For instance, in the example of estimating variance below, .
Multivariate case
Extending the Cramér–Rao bound to multiple parameters, define a parameter column vector
with probability density function which satisfies the two regularity conditions below.
Let be an estimator of any vector function of parameters, , and denote its expectation vector by . The Cramér–Rao bound then states that the covariance matrix of satisfies
,
where
The matrix inequality is understood to mean that the matrix is positive semidefinite, and
If is an unbiased estimator of (i.e., ), then the Cramér–Rao bound reduces to
If it is inconvenient to compute the inverse of the Fisher information matrix,
then one can simply take the reciprocal of the corresponding diagonal element
to find a (possibly loose) lower bound.[16]
The Fisher information is always defined; equivalently, for all such that , exists, and is finite.
The operations of integration with respect to and differentiation with respect to can be interchanged in the expectation of ; that is, whenever the right-hand side is finite. This condition can often be confirmed by using the fact that integration and differentiation can be swapped when either of the following cases hold:
The function has bounded support in , and the bounds do not depend on ;
The function has infinite support, is continuously differentiable, and the integral converges uniformly for all .
It suffices to prove this for scalar case, with taking values in . Because for general , we can take any , then defining , the scalar case gives This holds for all , so we can concludeThe scalar case states that with .
Let be an infinitesimal, then for any , taking in the single-variate Chapman–Robbins bound gives
.
By linear algebra, for any positive-definite matrix , thus we obtain
For example, let be a sample of independent observations with unknown mean and known variance .
Then the Fisher information is a scalar given by
and so the Cramér–Rao bound is
Normal variance with known mean
Suppose X is a normally distributed random variable with known mean and unknown variance . Consider the following statistic:
Then T is unbiased for , as . What is the variance of T?
(the second equality follows directly from the definition of variance). The first term is the fourth moment about the mean and has value ; the second is the square of the variance, or .
Thus
where the second equality is from elementary calculus. Thus, the information in a single observation is just minus the expectation of the derivative of , or
Thus the information in a sample of independent observations is just times this, or
The Cramér–Rao bound states that
In this case, the inequality is saturated (equality is achieved), showing that the estimator is efficient.
However, we can achieve a lower mean squared error using a biased estimator. The estimator
obviously has a smaller variance, which is in fact
Its bias is
so its mean squared error is
which is less than what unbiased estimators can achieve according to the Cramér–Rao bound.
When the mean is not known, the minimum mean squared error estimate of the variance of a sample from Gaussian distribution is achieved by dividing by , rather than or .
Fréchet, Maurice (1943). "Sur l'extension de certaines évaluations statistiques au cas de petits échantillons". Rev. Inst. Int. Statist. 11 (3/4): 182–205. doi:10.2307/1401114. JSTOR1401114.
Darmois, Georges (1945). "Sur les limites de la dispersion de certaines estimations". Rev. Int. Inst. Statist. 13 (1/4): 9–15. doi:10.2307/1400974. JSTOR1400974.